Question

A circular coil of radius \( R \) is carrying a current \( I_1 \) in anticlockwise sense. A long straight wire is carrying current \( I_2 \) in the negative direction of x-axis. Both are placed in the same plane and the distance between the centre of coil and straight wire is \( d \). The magnetic field at the centre of coil will be zero for the value of \( d \) equal to

• A. \( \frac{\pi}{R} \left| \frac{I_1}{I_2} \right| \)
• B. \( \frac{\pi}{R} \left| \frac{I_2}{I_1} \right| \)
• C. \( \frac{R}{\pi} \left| \frac{I_2}{I_1} \right| \)
• D. \( \frac{R}{\pi} \left| \frac{I_1}{I_2} \right| \)

Hint:

The magnetic fields due to a coil and a straight wire can cancel each other out when their magnitudes are equal. Use the Biot-Savart law for calculations.

Answer 1

The Correct Option is C

Step 1: Understanding the magnetic field contributions.
The magnetic field at the centre of the coil is due to the current flowing through the coil, and the magnetic field due to the long straight wire depends on the distance \( d \). For the magnetic fields to cancel each other, their magnitudes must be equal and opposite.
Step 2: Magnetic field due to the coil and the wire.
The magnetic field at the centre of the coil is: \[ B_{\text{coil}} = \frac{\mu_0 I_1}{2R} \] The magnetic field due to the wire at the distance \( d \) is: \[ B_{\text{wire}} = \frac{\mu_0 I_2}{2 \pi d} \]
Step 3: Condition for cancellation.
For the magnetic fields to cancel out: \[ \frac{\mu_0 I_1}{2R} = \frac{\mu_0 I_2}{2 \pi d} \] Solving for \( d \), we get: \[ d = \frac{R}{\pi} \left| \frac{I_2}{I_1} \right| \]
Step 4: Conclusion.
The distance \( d \) is \( \frac{R}{\pi} \left| \frac{I_2}{I_1} \right| \), so the correct answer is (C).